1

To appear in Journal of the Philosophy of Science Epistemic Groundings of Abstraction and Their Cognitive Dimension Sergio F. Martínez and Xiang Huang*+ ABSTRACT: In the philosophy of science, abstraction has usually been analyzed in terms of the interface between our experience and the design of our concepts. The often implicit assumption here is that such interface has a definite identifiable and universalizable structure, determining the epistemic correctness of any abstraction. Our claim is that, on the contrary, the epistemic grounding of abstraction should not be reduced to the structural norms of such interface, but is also related to the constraints on the cognitive processes of specific abstractions. This suggests that we should understand abstraction as embodied in different kinds of abstraction practices.

+ Contact information:

Sergio F. Martínez, Instituto de Investigaciones Filosóficas, Universidad

Nacional Autónoma de Mexico, Circ. Mario de la Cueva, C.U. Coyoacan,

Mexico D.F. 04510, Mexico; e-mail: sfmar@filosoficas.unam.mx. Xiang

Huang, School of Philosophy, Fudan University 220 Handan Road, Shanghai,

200433, P.R.China., email: hangxiang@fudan.edu.cn. Previously at Centro de

Investigaciones y Estudios Avanzados del Instituto Politécnico Nacional,

I.P.N., Mexico D.F.

* Previous versions of this article were presented at the University of Chicago

and the Universidad Autónoma Metropolitana, Thanks to the members of

those audiences. In particular we thank Axel Barcelo, Bill Wimsatt and

Rasmus Winther for their comments and discussions of the ideas here

presented.

2

1. Introduction. Abstraction is related to the way we learn from experience

and to the way we craft concepts in order to achieve goals. The epistemic

problem of abstraction consists in finding a criterion to allow us to decide

which abstractions are well grounded epistemically, and which are not. During

the 20th century, the predominant tradition in the analytic philosophy of

science sought to reduce the epistemic problem of abstraction to the study of

the way in which the conceptual structure of science could be exhaustively

analyzed as a logical structure. Well known accounts of abstraction usually

emphasize the processes of adding or subtracting properties to representations

that lend itself to such sort of analysis.1 Martin Jones introduces a distinction

between abstraction and idealization in the following way: “the term

‘idealization’ applies, first and foremost, to specific respects in which a given

representation misrepresents, whereas the term ‘abstraction’ applies to mere

omissions” (Jones 2005, 173). However, even if we concede the significance

of the distinction between misrepresentation and mere omission in

understanding how abstractions and idealizations operate in some important

sorts of models and laws, there are important cases in which we learn from

experience for which such distinction cannot be the whole story. For example,

Sarah de Rijcke (2008) shows that Santiago Ramón y Cajal’s abstraction of a

1 The kind of philosophical analyses carried out by people like Michael Friedman and Bas C. van Fraassen, who have been able to show how mathematics allow us to account for many important aspects of abstraction in mathematically structured scientific theories, is an impressive achievement (Friedman 1983, van Fraassen 1980, Pincock 2007, Jones 2005). Nonetheless, we argue against the risk of thinking that the epistemic problem of abstraction can be treated solely by recourse to mathematical or logical analysis.

3

nerve cell was not simply a process of omitting certain features of it. Rather, it

involved complex processes of scaffolded interaction between developing

skills for drawing and employing other techniques for visualizing the nerves

and the corresponding development of histological concepts. In such cases

understanding the epistemic dimension of abstractions requires going beyond

identification of what is omitted and what is misrepresented. Here, abstraction

is a process that has to be understood in terms of the deployment and

interaction of cognitive capacities and sorts of interaction with the world. In

this paper we are putting forward two related theses. First, we motivate our

conviction that the logicist tendency of reducing abstraction to the search for

criteria that can guide the adding or subtracting of properties to

representations in all domains of inquiry cannot exhaust the discussion about

the epistemic dimension of abstraction. We argue that the epistemic role of

abstraction cannot be reduced to analysis of concepts. There are different

types of abstraction that presuppose different kinds of epistemic criterion. In

sections 2 to 4, we introduce three contemporary theories of abstraction in

order to show that, even though they can be thought as competitors, the best

way to see them is as accounts of different kinds of abstraction not reducible

to each other. In section 2, we introduce Hans Radder’s and Nancy

Cartwright’s theories of abstraction. In Radder’s theory, the extensibility of

the concepts abstracted in practices of observation plays an important

epistemic role, whereas Cartwright’s theory proposes that the epistemic

criterion should be established within a complex interaction that takes place

4

between, on the one hand, assumptions of nature’s capacities, and, on the

other, practices of subtraction and concretization in the research project to

identify the causal mechanisms involved in the phenomena under

investigation. In section 4 we introduce Nersessian’s account of abstraction in

which the main epistemic weight comes from the power of analogical thinking.

Our claim is not that these different accounts of abstraction (and many others)

are just partial accounts that could be fit together like pieces of a jigsaw

puzzle to give us a more general theory of abstraction. Instead, such accounts

exemplify the different ways in which cognitive capacities (embodied in

instruments, routines, heuristics, institutions, etc.) interact with our methods of

learning from experience to generate grounded generalizations. This is not a

smooth process – there are overlapping claims that often remain in tension

with one another. However, the effect of those tensions goes beyond the scope

of our thesis.

The second thesis we want to defend will be presented in the section 5.

We argue that, understood as processes of grounded generalization, different

kinds of abstraction practices can be identified by the different sorts of

grounding. Such grounding can best be understood as constitutive of learning

processes in which different cognitive capacities are deployed in different

degrees. In this sense, the epistemic grounding of abstraction has a cognitive

dimension which cannot be bracketed by appealing to traditional distinctions

like that between discovery/justification, or by appealing to a sharp qualitative

distinction between the psychological (and more generally cognitive) aspects

5

of inquiry and the forms of explanations that matters in the philosophy of

science. We suggest that generalizations supported by abstraction are, a)

grounded in the different cognitive resources deployed in different research

projects, and b), are sensitive to the particular goals directing these projects.

2. Radder and Cartwright on abstraction. Hans Radder starts by

characterizing the classical doctrine of abstraction as ‘leaving out the

irrelevant particularities or idiosyncrasies” and “mentally setting apart what is

relevant and common” (Radder 2006, 110). This leads the inquirer from

individual spatio-temporal instantiations to a general concept, which then can

be taken to represent all the particular situations of a certain kind.

Radder argues that the classical doctrine of abstraction is wrong for the

following two reasons: first, it is not plausible to presuppose that some sort of

non-conceptual observation can be taken as the starting point of abstraction;

and second, the grounding of an abstracted concept depends not only upon the

appropriate actions of setting apart and leaving out, but also upon the

concept’s extensibility.

In his book of 2006 Radder focuses in developing the notion of

abstraction that is implied in the intended extension of concepts. An extensible

concept points to some set of realizability conditions that (if concretely

realized in actual space and time) could extend the concept to a specific new

domain. Extensible concepts have a nonlocal meaning (transcending the

6

meaning they have as interpretations of observational processes that have been

realized so far). Thus, extensible concepts do not have a fixed extension.

For Radder the well-known Kantian idea that concepts structure the

world is only part of the story. To the extent that concepts abstract from

particular domains in the process of being extended to a new realization

context the structuring meaning component of the concept shifts. Abstraction

is the process through which we set apart the result of a particular type of

process in order to use such result as scaffolding for further structuring2. Thus,

such account does not lead to a separation between what is abstract in

opposition to what is concrete, a rather usual view of abstraction, but it

requires the recognition that human independent potentialities, extensible

concepts with their nonlocal meanings, and local realizations of particular

observational processes constitute “sui generis ontological categories that

cannot be reduced to each other” (Radder 2006, 115).

Nor such account of abstraction is meant to be distinctive of scientific

concepts or “abstract” science, it applies to ordinary as well as to rather

esoteric scientific concepts. Abstraction as scaffolded potential realization in a

new context is a rather fundamental and irreducible aspect of our cognitive

life.

2 In 1996 Radder suggests a more general account of abstraction (not limited to extensible concepts) than the one presented in 2006. To abstract is an activity that, “being rooted in a fundamental indeterminacy that is inherent in the results of experimental processes, reflects a sensitivity to as yet unrealized possibilities” (Radder 1996, 85). In this book Radder talks of different kinds of replication procedures and suggest a pluralism about abstraction that is compatible to the sort of pluralism we propose in this paper. The different kinds of abstraction would be associated to different types of “stabilizing procedures”. But the elaboration of this point goes beyond our present concern.

7

We will end this brief account of Radder’s account of abstraction

pointing out the importance that a given notion of contingency plays in his

characterization of the extensibility of concepts. Concepts are not extensible

in any arbitrary direction. In advance of any attempted extension “it remains

fully contingent whether or not, in actual practice, a concept can and will be

successfully extended to a specific new domain”. Radder makes clear that

contingency here is used in its “common philosophical meaning of not being

(logically or physically) necessary, rather that in its common sense meaning of

being accidental or arbitrary” (Radder 2006, 103). It is this notion of

contingency that supports the extensibility of the concepts and the implied

notion of abstraction.

Nancy Cartwright (1989, 1999)’s theory of abstraction comes directly

from the well-known move in the philosophy of science, which counters the

empiricist emphasis on the law-like statements of regularities in science with

the claim that these are not enough to offer a satisfactory theory of scientific

explanation. Cartwright insists that regularities come into the picture only

after presupposing ascription of tendencies (where the claim is about what

things do) or of capacities (where the claim is about what things cause). It is

the latter two that do all the work in non-ad hoc explanations. “For example,

aspirins —because of being aspirins—can cure headaches. The troublesome

phrase ‘because of being aspirins’ is put there to indicate that the claim is

meant to express a fact about properties and not about individuals: the

property of being an aspirin carries with it the capacity to cure headaches”

8

(Cartwright 1989, 141). Cartwright’s notion of capacity and tendency come

respectively from Aristotle and Mill. In order to understand the nature of the

causal relations into which we are inquiring, we must isolate a particular

causal relation from other related causal relations to determine its tendency

and to measure its capacity. Scientists usually distinguish two kinds of causal

truth. At the high level of generality, the ascription of capacities assumes that

a causal relation between the cause A and the effect X occurs in every

homogeneous background. At the low level, there are causal laws that

articulate the causal relation between A and X in a population determined by

experiments.

Corresponding to the two levels of causal truth there are two kinds of

law-like statements about causation. The ascription of capacities corresponds

to the notion of abstract laws “A’s do X”, which expresses the general regular

association between A and X without taking into consideration any particular

concrete context of realization. They report what a particular cause A tends to

do in ideal circumstances if it has been isolated from other causal factors, but

these reports are not true or false in the sense that they do not literally describe

the behavior of real material systems. This is due to the fact that in a real

material system, a cause A can rarely be disentangled from other related

factors. But, when concrete realization context I is specified in experiments,

causal laws expressed as, “in I, A’s do X”, offer causal explanations that tell

us what A tends to do in a concrete experimental situation, controlling for all

9

related factors. Nature’s capacities are detected in terms of the causal laws,

together with the presupposition about the stability and regularity of nature.

Placed in this context, abstraction is related to the process by which

causal laws are obtained. Abstraction is not purely a construction of general

statements resulting from leaving out, setting apart and testing in other

experimental situations, but a constructive process that presupposes the notion

of capacities. More concretely, according to Cartwright, abstraction is

characterized by the following two kinds of processes: first, the process of

subtraction to obtain the abstract law of the form “A’s do X”, in which the

causal relation between the cause A and the effect X is isolated from other

related causal factors; and second, the process of concretization to obtain the

causal law of the form “In I, A’s do X”, in which the causal relation between A

and X is manifested in a particular condition I, determined many times by

concrete experiments. When abstract laws and causal laws use idealized

concepts or idealized models, their abstraction is a process of idealization.3

One example of these two processes of abstraction comes from Leszek Nowak

(1980)’s interpretation of Karl Marx’s law of value, which claims that the

market prices of commodities correspond with their value of production. This

an idealized abstract law subtracted from the reality of particular marketplace

situations. In the subtraction, many factors that affect real market, such as the

competition for producing and selling commodities, the difference between

3 This is a simplistic way of characterizing Cartwright’s notion of idealization. We use this way in order to avoid the complicate issue of differences between idealization and abstraction according to different authors (see Coniglione (2004) and Jones (2005) for reference).

10

the exports and the imports of a given economic system, the difference

between the average agricultural capitals and the average remaining capitals in

a given economic system, the merchant profit, etc., are bracketed off. Once

the abstract law of value is subtracted, what we have is an idealized relation

between a commodity’s value of production and its prices. However, although

this abstract law is supposed to indicate a causal tendency in an idealized

situation, it does not and is not meant to describe what happens in real

economical practices. In order to describe the real economical practices, Marx

searched for local causal laws by concretization, or by adding back the factors

omitted by the above presuppositions. It is by analyzing the local causal laws

that Marx was able to criticize the capitalist economy and society (Nowak

1980; Cartwright 1989, 203-205).

3. Radder’s criticism of Cartwright’s approach. Radder argues that

Cartwright’s theory of abstraction is not sustainable for the following two

reasons:

A1) Cartwright’s Aristotelian account of abstraction, “conceived as a method

of acquiring concepts or forming theories, is untenable because of the fact that

concepts and theories structure (our experience of) the world” (2006, 141);

11

A2) Cartwright’s concretization is not sustainable because, firstly, it is trivial

due to the fact that it is the inverse process of leaving out; and, secondly,

Cartwright here does not take in consideration that the extensibility of

concepts is what makes them abstract.

We believe that both reasons are questionable. Let’s look at A1) first.

A1) takes Cartwright’s use of the Aristotelian notion of abstraction as

evidence for her being a defender of the classical doctrine of abstraction,

according to which abstraction begins from some “direct or non-conceptual

observation of a number of particular situations”. As we have seen, Radder

argues that the classical doctrine is untenable because observation is always

conceptually interpreted.

However, even though Cartwright takes her inspiration from Aristotle,

it is not correct to say that she takes for granted non-conceptual observation as

the starting point of abstraction. Cartwright’s aim is not the defense of the

classical doctrine but, instead, the rejection of the idea that Humean regularity

statements are sufficient for scientific explanation. The abstraction of causal

law based on subtraction and concretization needn’t presuppose a non-

conceptual observation. In fact, it can easily be made compatible with

Radder’s Kantian position vis-à-vis scientific observation. As Cartwright says:

“[e]ven what are supposed to be the ‘purest’ empirical assertions, like ‘this

facing surface is red’, employ concepts which cannot be given ostensively but

only make sense relative to an entire structure of other concepts in which they

12

are embedded” (1989, 180). The difference between Cartwright and Radder

does not concern the Kantian notion of observation, but rather concerns their

commitments to different sorts of epistemic criterion.

As we have seen, Radder embraces a consequentialist methodology

according to which, if the process of generalization is inevitably interpreted by

concepts, its epistemic criterion and empirical contents should come from the

extensibility in other type of domains. For Cartwright, as also for Mill,

nature’s capacities or causal tendencies are presupposed in subtraction and

concretization, and thus, once detected and manipulated in experiments they

can be relied upon as an epistemic resource to justify the result of abstraction.

The ideal model of the epistemic criterion for abstraction adopted by

Cartwright is Mill’s “mixed method of induction and ratiocination”. It is the

process of “‘going upwards’ from experience to general principle and

‘argu[ing] downwards from that general principle to a variety of specific

conclusions’” (Cartwright 1989, 183). That is, subtraction aims to set apart a

general statement about the capacity, or the tendency, of a particular causal

factor, and to leave out any other potentially disturbing causal factors as

specified in terms of ceteris paribus conditions. Cartwright’s (and Mill’s)

going-upwards-step doesn’t suffice by itself to obtain abstract laws with

explanatory and predictive power, and thus must be paired with concretization

to generate relevant explanations and predictions. That is Cartwright’s version

of Millian ratiocination: some disturbing causal factors, isolated in the

13

inductive process, are then added back to the concrete material situation in

which the capacity of the considered causal factor manifests itself.

Still, Radder is not alone in finding Cartwright and Mill’s two step

abstraction suspiciously pre-Kantian. Humphreys, too, has expressed doubts

about whether there is a plausible epistemic account for how abstract laws are

subtracted from particulars that can avoid recourse to Aristotelian non-

conceptual observation (Humphreys 1995, 158-59). Let’s suppose that

subtraction or Millian induction is a process of Aristotelian abstraction as

leaving out, in which, “we begin with a concrete particular complete with all

its properties” or “[w]e then strip away—in our imagination—all that is

irrelevant to the concerns of the moment to focus on some single property or

set of properties” (Cartwright 1989, 197). The question for Mill and

Cartwright seems to turn on the ‘irrelevant’ – for if we do not assume non-

conceptual observation as our criterion here, how can we justify our

subtraction on epistemic grounds? For Radder, and perhaps for Humphreys, a

consequentialist explanation relying on the extensibility of abstracted concepts

is more plausible: in this way, we do not have to assign any particular

epistemic weight to the process of subtraction. For Cartwright, however, it is

not the metaphysically “essential properties” but rather some stable causal

mechanism that subtraction aims to grasp. Similar to Mill’s insistence that

scientists’ cognitive competency can offer sufficient justification for

abstraction practices, Cartwright suggests that ontological presuppositions

about the presence of nature’s capacities, together with experimental

14

manipulation aiming to establish the reliability of such presuppositions, can

serve as the epistemic ground of subtraction. Capacities of nature, material

and social conditions should be considered as the satisfactory epistemic

resources necessary to support the claim that a causal relation exists, even if

such statements are not infallible.

Radder’s A2 has two parts:

(A2.1) Concretization (as understood by Cartwright) does not takes in

consideration the extensibility of concepts; and

(A2.2) Concretization (as understood by Cartwright) is trivial as it is taken as

an inverse process of subtraction.

According to Cartwright, concretization is the process through which we

obtain those concrete causal laws that measure capacities through experiments

in a context in which certain interacting factors previously isolated by the

ceteris paribus conditions are put back in. In fact, Cartwright’s notion of

capacity is characterized by a kind of extensibility in as much as the

structures that represent a capacity should be persistent in different

environments. For example, Cartwright says: “To infer the stronger claim—

what I call a capacity claim—one must suppose that the causal possibilities

that are established in that situation continue to obtain in various different

kinds of situation” (1989, 147). In other places she associates this persistency

15

of capacities to the “the exportability of information”: “we gather information

in one set of circumstances but expect to use it in circumstances that are quite

different” (1989, 227). Like Radder, Cartwright assigns an epistemic weight

to the exportability of information, even though she doesn’t take it as the only

epistemic resource for grounding abstraction. With respect to the case of

concretization, a kind of extensibility is required to explain the relationship

between two interacting causal laws in different domains:

The first law gives evidence for a capacity, and that the capacity will

exhibit itself in a new causal law in any new test situation. That assumes

that capacity remains intact. It is, of course, part of the point of taking

capacities seriously as things in the world…that they should remain

intact from one kind of situation to another. But that does not mean that

there can be no exceptions; it means that any exception requires a reason.

Probably the most common reason for a capacity to fail to obtain in the

new situation is causal interaction (1989, 163).

Here, contrary to what Radder says, Cartwright doesn’t ignore extensibility.

She is characterizing a different kind of extensibility. Whereas Radder takes

the extensibility as the applicability of an abstract concept in a novel material

realization domain, for Cartwright, extensibility is a sign of a stable causal

relation allowing for prediction and manipulation. What this observation

suggests is that Cartwright and Radder are modeling different kinds of

16

abstraction practices, which are grounded in different notions of extensibility.

If Radder conceded the possibility of other kinds of abstraction that are not

based on the observation practices that he has characterized, then he would be

forced to grapple with Cartwright’s causal interpretation of extensibility. As

we shall see, our understanding of Radder and Cartwright’s different versions

of extensibility receives additional support from the discussion about the

triviality of concretization.

In concretization, or the process that leads from the top back down,

various of the factors taken away in subtraction are added back in, as we have

seen in Nowak’s example of Marx’s law of value. However, it is important to

point out that the conditions of subtraction are not the same as the conditions

of concretization. 4 As Cartwright says:

To get back to the concrete laws that constitute its phenomenal content,

the omitted factors must be added in again. But where do these omitted

factors come from? I have already described the answer I believe in:

given a theory, the factors come from a list. But the list provided by

any given theory, or even by all of our theories put together, will never

go far enough. There will always be further factors to consider which

4 Radder correctly points out that in cases of the replication of an observation through a radically different observational process, there are no omitted factors at all (2006, 101-102). Thus, we agree with Radder that there are types of abstraction that do not need concretization in the sense of Cartwright; hence, Cartwright’s model of abstraction should not be taken as describing all abstraction practices. However, what we aim to show in this paper is that 1) not all concretization is an inverse process of subtraction, and 2) concretization can be seen as a kind of extensibility.

17

are peculiar to the individual case. I call this the ‘problem of material

abstraction’. (1989, 206-7)

For Cartwright, material conditions are not mere contingent backgrounding

circumstances that the theoretical treatment of abstraction dispenses with. On

the contrary, when realized in a concrete experimental practice, they, together

with the theoretical components deriving from the research problematic,

specify the context in which one concretization is possible. For example, the

abstract law of the operation of bubble chamber is this: “a passing charged

particle has the capacity to cause bubbling in a liquid in a superheated state”

(1989, 209). However, before doing experiments, Donald Glaser did not have

any reason to think that the superheated media for his chamber could not be

xenon, but could be diethyl ether. This difference is decided by material

conditions of actual experiments in concretization, but it need not appear in

the list that represents the process of subtraction.

4. Nerssesian’s approach. The forms of concretization are thus inherently

very variable. In fact, many times concretization is not only not an inverse

process of subtraction but also a creative strategy of theory construction. This

can be seen in the example of James Clerk Maxwell’s construction of the

electromagnetic field studied by Nancy Nersessian (2002, 2008). Nersessian

and Cartwright share the same ultimate concern: how generalization functions

in and is derived from empirical knowledge. The difference is that, whereas in

18

Cartwright’s theory, abstraction aims to give the researcher a causal account

of phenomena, in Nersessian’s modeling theory, the aim of abstraction is to

obtain generic models that allow for a unified explanation of phenomena.

According to Nersessian, the generic model abstraction “is the process

of constructing a model that represents features common to a class of

phenomena” (2002, 129). She uses the process by which Maxwell developed

his mathematical theory of electromagnetic field in these terms: “… once

Maxwell formulated a satisfactory model representing a specific mechanism,

he considered those abstract relational structures of the mechanical model that

could account for the electromagnetic phenomena, formulated the equations of

the abstract model, and substituted in the electromagnetic variables”

(Nersessian 2008, 28). We can thus distinguish three steps in Maxwell’s

theoretical development. The first step can be characterized this way:

S1) In certain specific circumstances, experiments are performed whose

results fall under the scope of a local model expressed by law-like regularity

claims, very crudely of the form “In I, A’s do X” ( “I” being specified by the

ceteris paribus conditions of the experiments).

In our case, Nersessian foregrounds the visual version of the vortex-fluid

model that Maxwell constructed to account for the results of Michael

Faraday’s experiments concerning electromagnetic force. Roughly the model

can be expressed as “In an electromagnetic field specified by Faraday’s

19

experimental conditions (I), repulsive and attraction force manifested by

magnetic lines (A) can be modeled as vortex actions (X)”. In a series of

experiments performed from 1825 to 1831, Faraday observed the formation of

regular magnet lines in iron filings and interpreted this phenomenon to imply

that electromagnetic space is a force field of currents and charges. Maxwell

accepted this idea and suggested that the forces were transmitted through

Newtonian aether. In a paper published in 1855-6, Maxwell “provided a

kinematical analysis of magnetic lines of force as representing the intensity

and direction of the force at a point in space on analogy with the flow of an

imaginary, incompressible fluid through a fine tube of variable section”

(Nersessian 2002, 131). In a paper published in 1861-2, Maxwell specified

this imaginary fluid with the form “of centrifugal force of vortices in the

medium, with axes parallel to the lines of force” (2002, 132). This vortex

model should be established under a series of ceteris paribus conditions (I)

such as: “(1) electric and magnetic forces are at right angles to each other, (2)

magnetism is dipolar, and (3) the plane of polarized light passed through a

diamagnetic substance is rotated by magnetic action” (2002, 132).5

The visual model of the vortex is a “physical analogy” in which it is

assumed that the attractive and repulsive forces operate as stresses in a

mechanical aether. Given this hypothesis, one can assume that “relationships

that hold in the domain of continuum mechanics will hold in the domain of

5 In Nersessian (2008), I is characterized by a series of “constraints drawn from the target domain of electricity and magnetism, the source domains of continuum mechanics and machine mechanics, and from the constructed models themselves” (55).

20

electromagnetism” (2002, 136). We can interpret his analogical generic

modeling as a process of going through steps of subtraction to obtain a general

law-like statement, “A’s do X” from particulars.

The process to obtain the general laws from these models is the second

step:

S2) The statement without the restriction to the circumstances “A’s do X” is

subtracted.

In S2, “the mathematical representation of various magnetic phenomena is

derived by Maxwell from the vortex-fluid model”. Maxwell in this step “had

not yet specified a causal process in the aether connecting electricity and

magnetism, and so claimed not to have provided a mechanical explanation for

their interaction” (Nersessian 2002, 136). Maxwell’s aim is to obtain a generic

and idealized calculation system which didn’t necessarily have to be a

representation of real physical system. Like Cartwright’s similar use of

subtraction, in this step the abstract laws articulated mathematically are

obtained. However, in distinction from Cartwright’s use of subtraction, this is

not synonymous with the process of isolating a determinate causal relation

from all the other related causations manifest in singular causal situations.

In the third step, Nerssesian takes Maxwell to be undertaking a process

analogous to Cartwright’s concretization:

21

S3) The application of this statement “A’s do X” in explanation and prediction

is the method of concretization.

The purpose of concretization for Maxwell is to enable him to explain the

vortex motion in a physical system with (causally) extensible elements. 6

There is a problem of making the model real: if the attractive and repulsive

forces in electromagnetic field are stresses in a mechanical aether, then there

will be friction at the places of contact among vortices, which will lead to

jamming. The problem is solved by Maxwell with an interesting strategy: “by

supposing that a layer of particles, acting as idle wheels is interposed between

each vortex and the next, he stipulated that the particles would revolve in

place without slipping or touching in direction opposite to the vortices” (2002,

149). The invention of the notion of idle wheels seems ad hoc for eliminating

the consideration of friction between adjacent vortices. However, the

invention of the idle wheels “is consistent with the constraint that the lines of

force around a magnetic source can exist for an indefinite period of time, so

there can be no loss of energy in the model” (2002, 149). Maxwell thought

that, with the help of the idle wheels, the mathematical laws constructed in S2

can safely offer an explanation in the real physical system grounded in the

extensibility of some elements in the model. 6 Extensibility in this case can be identified with the sense of causal extensibility grounded on capacities we found in Cartwright’s account of abstraction, with a caveat. The interesting point (made to us clear by a referee) is that even though Maxwell is clear that the system he is instantiating need not exist in nature the kinds of causal relations between vortices and idle wheels support the extensibility of the physical system in question. In other words, the extensibility in question is not aggregative. The extensibility of parts of the system does not add up to the extensibility of the system as a whole. This sort of non-aggregative extensibility seems to be behind grounded generalizations supported by analogical reasoning.

22

The direct lesson we learn from Nersessian’s case study is this: it is

quite clear that in this case concretization is not inverse to the process of

subtraction. The construction of the analogy of idle wheels comes about

through concretization, which is organized to give a causal explanation of the

friction among moving vortices. They have not been considered in the earlier

steps S1 and S2, because the friction among the vortices is a question

appeared only after the abstract model of vortex had been invented through S1

and S2. The question is dependent on the invention and manipulation of an

abstract model in a particular problem-solving situation.This lead us to a

reconsideration of A2.2. Radder and Humpreys are correct in arguing that

concretization should not be understood as an inverse process of substraction.

However, concretization need not be understood in this way, as the cases of

Glaser and Maxwell show.

We hope that our line of argument by now makes it clear why

Radder’s criticism of Cartwright, even though it makes an important point,

overlooks a crucial feature of the problem under discussion, for contrary to

Radder’s premise, extensibility turns out to be a different sort of relation

under certain different circumstances. In fact, in Radder’s, Cartwright’s and

Nerssesian’s theory, abstraction’s extensibility is embodied in different

practices. The sort of extensibility on which Radder focuses is basically a kind

of aggregative extensibility, in which the epistemic role of extensibility

resides in its capacity of applying in other domains. This notion of

extensibility assumes that a causal law can be characterized by the

23

aggregation of the extensions of the law in the different concrete experimental

situations in which it is operative. Whereas in the sort of practices of

abstraction that Cartwright and Nersessian focuses on, the aim is to obtain a

causal explanation that often cannot be reduced to the sum of its extensions in

concrete experimental situations. In Cartwright’s and Nersessian’s sense of

abstraction, the epistemic resources configuring a process of abstraction

include not only the aggregative extensibility of a causal law, but also

domain-specific ontological considerations (that often lead to non-aggregative

extensibility). On Nersessian’s account, abstraction is closely related with

mental modeling and analogical inferences. Correct analogical inferences

characterize the appropriate extensions. As in Cartwright’s account, the

epistemic relevance of these types of extensibility cannot be satisfactorily

articulated merely in terms of logical structures and methodological rules.

Whether extensibility is aggregative or not is a question that has to be

answered depending on the situation at hand, and that requires that a model is

designed with the right sort of extensible elements.

Thus, it seems more appropriate not to talk of one relation of

extensibility but rather of different types of extensibility associated with

different presuppositions about how our interaction with the world has a role

in constructing and grounding what is ontologically the case. In the case of

Radder, our interactions with the world only reveal what is contingently there

(that for Radder means that our interactions reveal something that is not

logically necessary). But what our interactions with the world reveal to us can

24

be contingent in a different sense; in the case of Cartwright our interactions

with the world determine (fallibly) the causal-material contexts in which

objective causal relations can be identified; and, in the case of Nersessian, our

interactions with the world, in order to preserve their relevance in relation to

our inquiry, have to be mediated by epistemically relevant models. The

relevance in question passes through the identification of the right sort of

analogy. In other words, in the above three types of extensibility, different

ways of interaction with the world configure the different kinds of epistemic

criterion for abstraction (grounded in different types of contingently stable

interactions with the world). Our argument points to the fact that each of the

three types of abstraction we have examined individually can tell us only a

partial story of the epistemic dimension of abstraction; and, in particular, the

whole story about abstraction can’t be extracted from a theory that identifies it

with a shared logical structure.

Our reflection on Radder, Cartwright and Nersessian’s studies of

abstraction strongly suggests that, if the epistemic import of abstraction is

dependent on its grounding, then the shared logical structure among them

simply will not account for all epistemically relevant instances of abstraction.

That means that we have to pay attention to the way in which interactions with

the world give way to contingently stable extensions; such extensibility is

what makes them abstract (Radder’s insight).

25

5. On the cognitive dimension of abstraction. Our rejection of past and

current attempts to account for the epistemic relevance of all abstraction

processes in terms of a shared logical structure in favor of one that adopts a

pluralist attitude toward different ways to model abstraction practices receives

support from recent cognitive science research on abstraction that suggests

that different processes of abstraction are associated with different practices.

In our view, an epistemically and methodologically sound theory of

abstraction cannot be provided by a model that does not take into account of

the embeddedness of abstraction in cognition’s workflow. Different kinds of

abstraction are associated with different cognitive resources, and the

determination of what kind is relevant is given to us by the contextual factors

that configure different kinds of epistemic norms. In this section, we shall see

some examples of these studies. It is important to keep in mind that our aim is

not to show how the different accounts of abstraction in the philosophy of

science can be modeled by one specific cognitive model of abstraction. Rather,

our point is the following: looking at the variety of models and approaches to

abstraction in the cognitive sciences, it is evident that abstraction fills different

epistemic roles, shaped by different cognitive purposes.7 Thus, we get support

for the thesis that a plurality of abstraction practices is epistemologically

relevant in the philosophy of science.

7 Of course, our thesis relies on the assumption that abstraction as a psychological process is embodied or grounded, as such claim is defended by psychologists like the ones we discuss below. In particular see Bickhard (2001, 2009), Barsalou (2008), Anderson (2003) for arguments supporting (from different perspective) such point of departure.

26

It might be objected that the sort of abstraction processes involved in

doing science, and the sort of abstraction processes in children and common

folk addressed in the psychological literature are quite different, and thus that

the conclusion we want to draw is unwarranted. That is, we cannot extrapolate

from findings about the plurality of kinds of abstraction in psychology to the

plurality of kinds of abstraction in the philosophy of science. This objection

assumes a fundamental disjunction between the cognitive processes

underlying ordinary life and those underlying the sciences that we think is

untenable8.

Quite often psychologists have tried to understand the general features

of abstraction, such as generalization, creativity, problem solving capacity, etc,

by classifying different kinds of abstraction according to their cognitive

purposes, precisely for the reason that different cognitive purposes implies

different methodological (and sometimes epistemic) norms for abstraction.

This idea can be found in Jean Piaget’s study on the development of abstract

thought among children as they age. Piaget distinguishes between empirical

abstraction that is used to organize observed objects and reflective abstraction,

which is abstraction about concepts and actions (Piaget, 1968). The cognitive

structures of these two processes are different: in the first, the basic operation

is that of setting up correspondences between words and their signifieds,

8 Carruthers (2002) and Eraña and Martínez (2004) provide specific arguments for the continuity thesis independently of claims about abstraction. Radder, Cartwright and Nersessian support versions of the continuity thesis as part of their accounts of the sort of abstraction that matters in science. Extensibility for these authors is a general feature of human practice.

27

whereas in the second, the coordination of actions serves as the basis for the

higher order cognitive acts (for instance, taking empirical abstractions as an

object of further abstraction) .

Colunga and Smith (2003) offer a more sophisticated explanation of

this differences based on the results of strictly controlled experiments.

According to Colunga and Smith, after a brief period in which children learn

instance-by-instance label-words, children begin to generalize the regular

pattern or structure of the objects observed through associative learning of

similarity. The generalized regularity is abstract in the sense that it is a rule-

like guide helping the children to apply it to instances never seen before. This

process is sensitive to different contexts, because the patterns of regularities

are characterized not only in terms of properties shared by the objects, but also

by a special way to classify or to partition objects. The abstract category of

solid things, for example, is related to the shape of things, whereas the notion

of non-solid thing is related to the material of which things are composed.

Therefore, children “become learners who have partitioned the learning space

into fundamentally different kinds of problems—learning about words,

learning about animal sounds, learning about objects, learning about

substances” (p. 1213). For Colunga and Smith, the effect of these partitions is

to sensitize our manipulations of mental concepts to various contexts

associated with different cognitive purposes. The relations between contexts

and purposes not only provides a grounding for abstraction, but it leads us to a

pluralist view of abstraction in order to understand how different sorts of

28

learning processes are related to different sorts of partition, which entail, in

turn, different sorts of abstraction.

In cognitive linguistics there are several proposals exploring the idea

that abstract concepts are understood through metaphors. For Lakoff and

Johnson (1999, ch.5) metaphors like “time flies” should be seen as mappings

of a physical experience of spatial movement that gives us a figurative

understanding of time. Alternatively, Lera Boroditsky (2000) claims that

people’s thinking about “the abstract domain of time” is built on

representations of more experience-based domains, but not on the physical

experience itself. This suggests that abstraction may occur differently at

different cognitive levels, and one has to understand those levels. If abstract

thinking about time should be understood in relation to spatial thinking (and

not necessarily in relation to the experience of motion itself) then our idea of

causality may well change depending on the way we think of the relation

between time and space. Following this train of thought, it seems natural to

think that different views of abstraction depend on the different cognitive

purposes we might have in a given situation. In the work of Lakoff, Johnson

and Boroditsky, monism about abstraction clearly leads us to the wrong

conclusions.

From a different perspective, Lawrence Barsalou takes a similar stance

about the plurality of processes of abstraction. According to Barsalou

abstraction is a cognitive mechanism in which “association areas in the brain

capture modality-specific states during perception and action, and then

29

reinstate them later to represent knowledge”. For example, “during visual

processing of a car, populations of neurons fire for edges, vertices and planar

surfaces, whereas others fire for orientation, color and movement. The total

pattern of activation over this hierarchically organized distributed system

represents the entity in vision” (2003, 1179). Two stages are involved in this

mechanism: one in which the modality-specific patterns are activated through

perception and action and stored in memory; and then the re-enactment stage,

in which associated neurons “later reinstate the pattern in the absence of

bottom-up stimulation”(2003, 1180). The application of the re-enactment

stage can be taken as a simulation of what the original modality-specific stage

delivers, that is, simulators “integrate information across a category’s

instances”, and simulations “are specific conceptualizations of the category”

(2009, 1282). The concrete abstraction realized by a particular simulator,

however, only partially determines the content of an abstracted concept, as a

concept can be re-enacted by different simulators in response to different

contexts. As Barsalou says, “on one occasion, the car simulator might

produced a simulation of traveling in a car, whereas on others it might

produce simulations of repairing a car, seeing a car park and so forth” (2003,

1180). Corresponding to distinct action-situations such as driving, washing or

repairing, different simulators endow the concept of car with different

properties structured by different types of relations. A simulation realized by a

determinate simulator in this sense is holistic and contextual in character.

30

The fact that many simulations determine an abstracted concept makes

an abstraction dynamic in the sense that “although a relatively fixed set of

property and relation simulators may exist for a person at a give point in time,

the particular ones used across occasions vary considerably” (2003, 1183).

What triggers a simulator and not others is determined by such exiguous

factors as frequency and recentness: “some simulators may be more likely to

be applied than others. Simulators applied frequently in the past will have an

advantage, as will simulators applied recently” (2003, 1183). The point that

we want to emphasize is that extensibility of a simulation depends on the

simulator. Thus, Barsalou can be taken as suggesting a way in which the

different types of extensibility can be cognitively grounded.

The dynamic character of abstraction implies that searching for a

unique theory of abstraction is pointless, since abstraction processes must

function differently relative to cognitive purposes. As Barsalou claims, “no

one abstraction can be identified and motivated as the summary representation

of a category, because an infinite number are possible” (2003, 1183).

Epistemic norms are thus often needed to determine which simulation should

be triggered. And the role of norms will depend on the type of abstraction that

a situation or context demands.

Barsalou proposes a list of six types of abstraction investigated by

cognitive scientists, as follows:

31

1) Categorical knowledge - the process of relating appropriately a concept

with the term of it, e.g., the concept of chair with the term “chair” and not

with “table” or “cat”.

2) The behavioural ability to generalize across category members, which is an

abstraction of the knowledge about the behavioural properties of a category,

e.g. “dogs bark”.

3) Summary representation - generalization of things such as a declarative rule,

a statistical prototype or a connective relation.

4) Schematic representation - abstraction of the critical relevant properties of a

category’s exemplars, discarding any irrelevant feature. Schema may be

distorted to idealize or caricature a category.

5) Flexible representation – the application of summary representation to

many different tasks, including categorization, inference, language

comprehension, reasoning, etc.

6) Abstract concepts, which is the abstraction of the abstract concepts such as

“pleasure”, “courage”, etc (2003, 1185; 2005, 424-25).

Radder, Cartwright and Nersessian point to different ways in which

different processes of abstraction can enter into scientific concepts and

practices. For example, Radder’s theory of abstraction could be explained

against the background of Barsalou’s approach in the following way: practices

of finding the regular pattern in a set of observed objects and extending this

pattern to other areas of observation are realized through modality-specific

mechanisms associated with categorical knowledge, summary representation,

32

or schematic representation. In these cognitive processes, specific simulators

are established in order to let the subject observe the objects in a determinate

way. Besides, the re-enactment process specifies extensibility of an abstracted

concept, which should be understood as a kind of flexible representation, in

which “summary representations can be applied flexibly to many different

tasks” (2003, 1185). It is a step related to the subject’s background knowledge

and skill more than to her capacity of capturing the correct description of a

concept.

Quite differently, Cartwright insight that causal relations are based on

nature’s capacities is supported by the extensibility of behaviors, the tendency

of such behaviors to generalize across category members. Causal explanations

such as “aspirins relieve headaches” or “electromagnetic forces cause motions

perpendicular to the line of action” refer to nature’s capacities via the

behavioral tendencies of aspirins’ and electromagnetic forces’. In Barsalou

terminology, such causal relations are abstract products of simulators for

aspirin and of electromagnetic force, developed through modality-specific

mechanisms that will allow us to interpret a particular headache and a

particular motion perpendicular to the line of action as result of such causes.

As we have seen, Cartwright distinguishes between the abstract law

articulated as “A’s do X” and the causal law articulated as “In I, A’s do X”.

The first is a general law about regularity subtracted from experimental data,

whereas the second is the local causal law about a capacity of nature obtained

by concretization. From Barsalou’s perspective, the abstract law can be

33

characterized by summary and schematic representations, in which a regular

pattern is generalized from experimental data. Whereas the local causal laws

are results of flexible representation, in which a concrete application domain I

is detected by skillful scientists in experiments.

In Nersessian’s generalization by analogy, besides summary representation,

schematic representation and abstract concepts involved in the construction of a

model, flexible representation plays the special role of transferring some regularity

from one kind of phenomena to another kind of phenomena through analogy.

Nersessian is calling attention to the importance to generalizations formulated in

terms of generic abstraction that cannot be characterized merely as an application of

some regular rules in new domains, but has to be characterized as a creative process

involving the manipulation and modification of the original regular patterns in such a

way that a new “natural kind” emerges as part of the objective phenomena science

deals with. In Barsalou’s words, “when the same configuration of property and

relation simulators can be applied to different categories, analogy becomes possible”

(2005, 422). The generalization by analogy might be explained as the core of what

Barsalou calls flexible representation. Flexible representations are not merely a re-

enactment simulation of those modality-specific patterns triggered by the original

contextual simulators, as in the case of prediction. In an analogical representation the

re-enactment simulation is not the inference of a conceptualization in a similar

situation to the one in which this conceptualization is initially activated, but is a

creative inference involving the identification of a different kind of phenomena, a

more abstract kind. In Maxwell’s case, patterns and regularities originally found in

34

phenomena involving fluids and electrostatics were transferred into a new kind of

phenomena (electromagnetic ones). According to Barsalou, the successful

establishment of this creative step depends on whether the newly transferred

conceptualized pattern can allows us to re-enact simulation applicable in the area of

electromagnetic phenomena. Maxwell’s invention of the idle wheels might be

considered as an attempt to construct suitable simulators in environments mediated by

aether, in order to make possible the re-enactment of the vortex-fluid model in future

prediction and explanation.

Understanding how different cognitive resources play a role in explaining

different sorts of epistemic grounding of abstraction promoted by different

philosophers requires more careful and systematic investigation. But this is outside of

the scope of this paper. Also, it is important to say explicitly that our thesis is not

committed to the specific way in which Barsalou models the plurality of abstraction

processes in terms of modality specific mechanisms. We consider it a plausible

explanation of the way in which the plurality of abstraction processes relevant in

scientific and everyday reasoning is grounded in cognitive resources.

However, we think that understanding the epistemic grounding of abstraction

requires going beyond an account like that of Barsalou, it requires introducing

concepts like that of affordance9. Affordances can be seen as (expression of) patterns

9 The concept of affordance was introduced by Gibson (1979) as the key concept of his ecological approach to perception. Gibson’s affordances allow us to say that meaning is not “inside” the perceiver, but rather that meaning is constitutive of affordances that are part of the environment. Affordances can be understood as abstract features of the environment pointing to possibilities of action. Affordances are then intrinsically relational (see Chemero 2009, Ch. 7). In this view affordances would then be an embodiment of behavioral abstractions, schematic and summary

35

of potential action (see for example Borghi 2000). For Borghi, affordances constitute

genealogies of ways of knowing how to do things that in the case of science are

usually articulated by scientific practices and theoretical concepts like that of

“electromagnetic field”. Different types of abstraction can be seen as different ways

of organizing the different sort of norms and cognitive resources that form the

entrenched patterns of activity that coalesce into a practice. From this perspective,

affordances capture abstract thinking as embodiments of ways of knowing how that

can be extrapolated into new domains of knowledge (assuming that such affordances

embody the right epistemic norms). Barsalou’s account is centred on the integration

of modality. Its specific processes might seem quite different from an account of

abstraction based on affordances, but they coincide in promoting a view of

abstraction according to which the relative weight and the genealogy of different

processes of abstraction scaffolds its epistemic function. To the extent that those

empirical investigations support this basic point, they are supporting our thesis.

Summing up, cognitive studies of abstraction give us good reason to adopt the

pluralist stance according to which different (epistemically relevant) types of

abstraction will always arise from different cognitive resources and epistemic norms,

and must be understood relative to such differences. Such pluralist approaches allow

us better to understand the way in which the interplay of different types of abstraction

representations at once. However, different affordances would embody different abstraction processes. For example, Cartwright’s capacities embody affordances implicit in an experimental set-up, Nersessian’s generic modeling points to abstraction processes embody in affordances that can be understood as flexible representations connecting different kinds of things.

36

processes plays a role in the generation of stable practices embodying a complex

variety of implicit and explicit epistemic norms.

References

Anderson, Michael L. 2003. “Embodied Cognition-A field Guide”, Artificial

Intelligence, 149: 91-130.

Barsalou, Lawrence W. 2003. “Abstraction in perceptual symbol systems” Phil.

Trans. R. Soc. Lond. B 358: 1177-1187.

-- 2005. “Abstraction as Dynamic Interpretation in Perceptual Symbol Systems” in

Building Object Categories, eds. Carnegie Symposium Series, L. Gershkoff-Stowe &

Rakison, 389-431. Majwah, NJ: Erlbaum..

--2008 “Grounded Cognition” Annual Review of Psychology, 59: 617-45.

-- 2009. “Simulation, situated conceptualization, and prediction” Phil. Trans. R. Soc.

B. 364: 1281-89.

Bickhard, Mark H. 2001. “Why Children Don’t Have to Solve the Frame Problems:

Cognitive Representations are not Encodings” Developmental Review, 21: 224-62.

--2009. “The Interactivist Model”, Synthese, 166: 547-91.

Borghi, Anna M. 2002 “Object Concepts and Action” in Grounding Cognition—The

Role of Perception and Action in Memory, Language, and Thinking, eds. Diane

Pecher and Rolf A. Zwaan, Cambridge: Cambridge University Press.

Boroditsky, Lera. 2000. “Metaphoric Structuring: Understanding Time through

Spatial Metaphors”, Cognition, 75: 1-28

37

Carruthers, Peter. 2002. “The roots of scientific reasoning: infancy, modularity, and

the art of tracking” in The Cognitive Basis of Science. eds. P. Carruthers, S. Stich and

M.Siegal, 73-96. Cambridge: Cambridge University Press.

Cartwright, Nancy D. 1989. Nature’s Capacities and their Measurement, Oxford:

Clarendon.

--1995. “Reply to Eells, Humphreys and Morrison” Philosophy and

Phenomenological Research, IV-1: 177-87.

--1999. The Dappled World—A Study of the Boundaries of Science, UK: Cambridge

University Press.

Chemero Anthony. 2009 Radical Embodied Cognitive Science, Cambridge Mass.

MIT Press.

Colunga, Eliana and Linda B. Smith. 2003. “The emergence of abstract ideas:

evidence from networks and babies” Phil. Trans. R. Soc. Lond. B358: 1205-14.

Coniglione, Francesco. 2004. “Between Abstraction and Idealization: Scientific

Practic and Philosophical Awareness” in eds. Coniglione, et. al., 59-110.

Coniglione, Francesco, Roberto Poli and Robin Rollinger. 2004. (eds.) Idealization XI:

Historical Studies on Abstraction and Idealization, Poznan Studies in the Philosophy

of the Sciences and the Humanities, vol. 82, Amsterdam, New York: Rodopi B.V.

de Rijcke, Sarah. 2008. “Drawing into Abstraction. Practices of observation and

visualization in the work of Santiago Ramón y Cajal” Interdisciplinary Science

Reviews, 33(4), 287-311.

Erana Angeles, Martínez Sergio. 2004. “The Heuristic Structure of Scientific

Knowledge”, Journal of Cognition and Culture, 4: 701-29.

38

van Fraassen, Bas. 1980. The Scientific Image, Oxford: Oxford University Press.

Friedman, Michael. 1983. Foundations of Space-Time Theories: Relativistic Physics

and the Philosophy of Science, Princeton University Press.

Gibson, James J. 1979. The Ecological Approach to Visual Perception, Boston:

Houghton Mifflin.

Humphreys, Paul. 1995. “Abstract and Concrete” Philosophy and Phenomenological

Research, IV: 157-61.

Jones, Martin R. 2005. “Idealization and Abstraction: A Framework” in eds. Jones

and Cartwright, 172-217.

Jones, Martin R and Nancy Cartwright. 2005. (eds.) Idealization XII: Correcting the

Model. Idealization and Abstraction in the Sciences (Poznan Studies in the

Philosophy of the Sciences and the Humanities, vol. 86, Amsterdam/New York, NY:

Rodopi.

Lakoff, George and Mark Johnson. 1999. Philosophy in the Flesh—The Embodied

Mind and Its Challenge to Western Thought, New York: Basic Books.

Nersessian, Nancy J. 2002. “Abstraction via Generic Modeling in Concept Formation

in Science” Mind &Society, 5-3: 129-54.

--2008. Creating Scientific Concepts, Cambridge, Massachusetts: The MIT Press

Nowak, Leszek. 1980. The Structure of Idealization—Towards A Systematic

Interpretation of The Marxian Idea of Science, Dordrecht: Reidel Publishing

Company.

Piaget, Jean. 1968. Genetic Epistemology, New York: Columbia University Press.

39

Pincock, Christopher. 2007. “Mathematical Idealization” Philosophy of Science, 74:

957-67.

Radder, Hans. 2006 The World Observed/The World Conceived, Pittsburgh:

University of Pittsburgh Press.

-- 1996 In and About the World Philosophical Studies of Science and Technology,

New York, State University of New York Press.